Pick the relation: C = C₀·e−kt, LD = Vd·C/(S·F), maintenance rate = CL·Css/(S·F), τ = ln(Cpeak/Ctrough)/k.
For kidney function, estimate CrCl with Cockcroft–Gault first, then scale the dose or the interval.
Check the result: levels fall by half every half-life, and steady state takes about five half-lives.
Pharmacy calculations
Percentage strength
% w/v = g per 100 mL
Amount in a solution
g = % × mL / 100
Ratio strength
1:X = 1 g in X mL; % = 100 / X
Reconstitution
final volume = diluent + powder volume
Concentration
mg/mL = drug in vial / final volume
Alligation
parts of H = T − L, parts of L = H − T
Water or base
0%
Pediatric & critical-care dosing
Daily dose
mg/day = mg/kg/day × weight (kg)
Doses per day
number = 24 h / dosing interval (h)
Pounds to kilograms
kg = lb / 2.2
Infusion concentration
units/mL = units in bag / bag volume
Pump rate
mL/h = dose per hour / concentration
Titrated drip
mL/h = mcg/kg/min × kg × 60 / (mcg/mL)
Dose from pump rate
mcg/kg/min = mL/h × (mcg/mL) / (60 × kg)
Units
1 mg = 1000 mcg
2Practice problems
1.Clinical pharmacokineticsMedium
Practice exercise (made-up values, not clinical guidance). A patient receives an IV bolus of 270 mg of gentamicin. The level back-extrapolated to time zero is 10 mg/L and the half-life is 3 h. Find the volume of distribution Vd.
D = 270 mgC₀ = 10 mg/Lt½ = 3 h
Vd = L
2.Clinical pharmacokineticsHard
Practice exercise (made-up values, not clinical guidance). A patient receives vancomycin IV every 12 h; the vancomycin clearance is 3 L/h. What dose D in mg gives an average steady-state level of 20 mg/L?
CL = 3 L/hτ = 12 hS = 1F = 1Css,avg = 20 mg/L
D = mg
3.Pharmacy calculationsMedium
Practice exercise (made-up values, not clinical guidance). A vial of penicillin G contains 5 000 000 units. Its practice label gives three strengths (see the table). You add 18.2 mL of diluent. The prescribed dose is 400 000 units. What volume V do you draw up? Round to one decimal place.
18.2 mL → 250 000 units/mL400 000 units
V = mL
4.Pharmacy calculationsHard
A vial holds 1000 mg of powder. You add 9.8 mL of diluent, and the powder adds 0.2 mL of volume. The prescription is 150 mg. Draw up the dose: what volume V do you give?
1000 mg9.8 mL + 0.2 mL150 mg
V = mL
5.Pediatric & critical-care dosingMedium
Practice exercise (made-up values, not clinical guidance). A bag contains 25 000 units of heparin in 500 mL. The order is 18 units/kg/h for a patient who weighs 87 kg. At what rate R do you set the pump, in mL/h? Round to one decimal place.
25 000 units/500 mL18 units/kg/h87 kg
R = mL/h
6.Pediatric & critical-care dosingHard
Practice exercise (made-up values, not clinical guidance). A dopamine infusion of 400 mg in 250 mL runs at 8 mL/h. The patient weighs 78 kg. What dose D in mcg/kg/min is the patient receiving? Round to one decimal place.
400 mg/250 mL8 mL/h78 kg
D = mcg/kg/min
7.PharmacologyMedium
What makes an overdose of tricyclic antidepressants such as amitriptyline especially dangerous?
ASudden high blood glucose
BCardiac arrhythmias and seizures
CKidney stones
DSevere liver failure within hours
8.PharmacologyHard
Why is strict aseptic technique and a limited hang time needed for propofol infusions?
APropofol becomes more potent the longer it hangs
BPropofol breaks down into a toxic gas after a few hours
CLight exposure changes propofol into an opioid
DIts lipid emulsion lets bacteria grow if it is contaminated
3Solutions
1.Clinical pharmacokinetics
Answer: Vd = 27 L
Clearance and volume: at steady state the dosing rate equals the elimination rate, so the maintenance rate = CL × Css. A loading dose is LD = Vd × C/(S × F), with S the salt factor (fraction of active drug in the salt) and F the bioavailability (1 for IV). From two levels, k = ln(C₁/C₂)/Δt; for an IV bolus Vd = dose/C₀ and CL = k × Vd. The numbers are for practice only; at the bedside always follow local protocols and have high-risk doses double-checked.
Volume of distribution
Vd = 270/10 = 27 L
Answer
Vd = 27 L
2.Clinical pharmacokinetics
Answer: D = 720 mg
Designing a regimen: with rapid IV doses the level falls from peak to trough during the interval, so τ = ln(Cpeak/Ctrough)/k and the dose that restores the peak is Vd × (Cpeak − Ctrough). The average steady-state level is Css,avg = S·F·dose/(CL·τ). An infusion rises as C = (R/CL)(1 − e−kt). For a drug cleared by the kidneys, the dose is scaled with the creatinine clearance. The numbers are for practice only; at the bedside always follow local protocols and have high-risk doses double-checked.
Dose
D = Css,avg·CL·τ/(S·F) = 20·3·12/(1·1) = 720 mg
Answer
D = 720 mg
3.Pharmacy calculations
Answer: V = 1.6 mL
Reconstitution: the powder takes up space, so the final volume is the diluent plus the powder volume. Concentration = drug in the vial/final volume, and the volume to give = dose/concentration. On a multi-strength label, use the concentration that belongs to the amount of diluent you actually added. The numbers are for practice only; at the bedside always follow local protocols and have high-risk doses double-checked.
Concentration from the label
18.2 mL → 250 000 units/mL
Volume to give
V = 400 000/250 000 = 1.6 mL
Answer
V = 1.6 mL
4.Pharmacy calculations
Answer: V = 1.5 mL
Strength to volume: turn the strength into mg per mL first. A w/v percentage is grams per 100 mL, so x% = 10x mg/mL. A ratio strength 1:n is 1 g in n mL. After reconstitution the concentration is the powder's dose over the final volume (diluent + powder volume). Then volume = dose/concentration.
Final volume and concentration
9.8 + 0.2 = 10 mL
C = 1000/10 = 100 mg/mL
Volume
V = 150/100 = 1.5 mL
Answer
V = 1.5 mL
5.Pediatric & critical-care dosing
Answer: R = 31.3 mL/h
Heparin and insulin infusions: concentration = units in the bag/bag volume (units/mL). Pump rate (mL/h) = units per hour/concentration, and units per hour = mL/h × concentration. For a weight-based order, units/h = units/kg/h × weight first. A bolus volume = units needed/vial strength. The numbers are for practice only; at the bedside always follow local protocols and have high-risk doses double-checked.
Units per hour
18·87 = 1 566 units/h
Concentration
25 000/500 = 50 units/mL
Pump rate
R = 1566/50 = 31.32 ≈ 31.3 mL/h
Answer
R = 31.3 mL/h
6.Pediatric & critical-care dosing
Answer: D = 2.7 mcg/kg/min
Titrated drips: first find the concentration in mcg/mL (mg × 1000/mL). For a weight-based order, mcg/min = mcg/kg/min × weight. Then mL/h = mcg/min × 60/concentration. Working backwards, mcg/kg/min = mL/h × concentration/(60 × weight). The numbers are for practice only; at the bedside always follow local protocols and have high-risk doses double-checked.
Concentration
400 000/250 = 1600 mcg/mL
Micrograms per minute
8·1600 = 12800 mcg/h
12800/60 = 213.3 mcg/min
Per kilogram
D = 213.3/78 = 2.735 ≈ 2.7 mcg/kg/min
Answer
D = 2.7 mcg/kg/min
7.Pharmacology
Answer: B · Cardiac arrhythmias and seizures
Why this is the answer
Tricyclics block cardiac sodium channels (widening the QRS complex) and have anticholinergic effects. In overdose they cause arrhythmias, hypotension, seizures and coma.
Answer
B · Cardiac arrhythmias and seizures
8.Pharmacology
Answer: D · Its lipid emulsion lets bacteria grow if it is contaminated
Why this is the answer
Propofol is formulated in a soybean oil emulsion, which is a good growth medium for bacteria. Contaminated propofol has caused sepsis, so it is handled aseptically and tubing and containers are changed at set intervals.
Answer
D · Its lipid emulsion lets bacteria grow if it is contaminated